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Potential: V = kQ/r (point charge). V = kp*cos(theta)/r^2 (dipole). V_ring = kQ/sqrt(x^2+R^2). V_sphere(inside solid) = kQ(3R^2-r^2)/(2R^3). V_sphere(inside hollow) = kQ/R.

E-V relation: E = -dV/dr. E = -grad(V). V_A - V_B = -integral from A to B of E.dr. For uniform field: V = -Ed + C.

Potential energy: U = kq1q2/r (two charges). U_system = sum over all pairs. W = q*delta(V). Number of pairs = N(N-1)/2.

Self-energy: Solid sphere: 3kQ^2/(5R). Shell: kQ^2/(2R).

Capacitance: C = Q/V. Parallel plate: C = epsilon_0A/d. With dielectric: C = Kepsilon_0A/d. Spherical: C = 4piepsilon_0ab/(b-a). Cylindrical: C = 2piepsilon_0L/ln(b/a). Isolated sphere: C = 4piepsilon_0R.

Combinations: Series: 1/C_eq = sum(1/C_i). Parallel: C_eq = sum(C_i). Two in series: C1*C2/(C1+C2).

Energy: U = (1/2)CV^2 = (1/2)QV = Q^2/(2C). Energy density: u = (1/2)epsilon_0E^2.

Dielectric (battery on): V same, C -> KC, Q -> KQ, U -> KU. Dielectric (battery off): Q same, C -> KC, V -> V/K, U -> U/K.

Redistribution: V_f = (C1V1 + C2V2)/(C1+C2). U_lost = C1C2(V1-V2)^2/(2(C1+C2)).

Force between plates: F = Q^2/(2epsilon_0A). Pressure: P = sigma^2/(2*epsilon_0).

Partial dielectric: Series model C = epsilon_0A/(d-t+t/K). Conducting slab: C = epsilon_0A/(d-t).

Key constants: k = 9 x 10^9 N m^2/C^2. epsilon_0 = 8.854 x 10^(-12) F/m. 1 eV = 1.6 x 10^(-19) J.